\(\int \frac {\log (c (d+e \sqrt {x})^p)}{f+g x^2} \, dx\) [266]

Optimal result

Integrand size = 24, antiderivative size = 541 \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \] Output:

-1/2*ln(c*(d+e*x^(1/2))^p)*ln(e*((-(-f)^(1/2))^(1/2)-g^(1/4)*x^(1/2))/(e*( 
-(-f)^(1/2))^(1/2)+d*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*ln(c*(d+e*x^(1/2))^p 
)*ln(e*((-f)^(1/4)-g^(1/4)*x^(1/2))/(e*(-f)^(1/4)+d*g^(1/4)))/(-f)^(1/2)/g 
^(1/2)-1/2*ln(c*(d+e*x^(1/2))^p)*ln(e*((-(-f)^(1/2))^(1/2)+g^(1/4)*x^(1/2) 
)/(e*(-(-f)^(1/2))^(1/2)-d*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*ln(c*(d+e*x^(1 
/2))^p)*ln(e*((-f)^(1/4)+g^(1/4)*x^(1/2))/(e*(-f)^(1/4)-d*g^(1/4)))/(-f)^( 
1/2)/g^(1/2)-1/2*p*polylog(2,-g^(1/4)*(d+e*x^(1/2))/(e*(-(-f)^(1/2))^(1/2) 
-d*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*p*polylog(2,-g^(1/4)*(d+e*x^(1/2))/(e* 
(-f)^(1/4)-d*g^(1/4)))/(-f)^(1/2)/g^(1/2)-1/2*p*polylog(2,g^(1/4)*(d+e*x^( 
1/2))/(e*(-(-f)^(1/2))^(1/2)+d*g^(1/4)))/(-f)^(1/2)/g^(1/2)+1/2*p*polylog( 
2,g^(1/4)*(d+e*x^(1/2))/(e*(-f)^(1/4)+d*g^(1/4)))/(-f)^(1/2)/g^(1/2)
 

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 0.31 (sec) , antiderivative size = 422, normalized size of antiderivative = 0.78 \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+i \sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-i d \sqrt [4]{g}}\right )+\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )+p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )-p \operatorname {PolyLog}\left (2,\frac {i \sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+i d \sqrt [4]{g}}\right )-p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{i e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )+p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}+d \sqrt [4]{g}}\right )}{2 \sqrt {-f} \sqrt {g}} \] Input:

Integrate[Log[c*(d + e*Sqrt[x])^p]/(f + g*x^2),x]
 

Output:

(Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^( 
1/4) + d*g^(1/4))] - Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) - I*g^(1/ 
4)*Sqrt[x]))/(e*(-f)^(1/4) + I*d*g^(1/4))] - Log[c*(d + e*Sqrt[x])^p]*Log[ 
(e*((-f)^(1/4) + I*g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - I*d*g^(1/4))] + Log[c 
*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) - 
 d*g^(1/4))] + p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) - d* 
g^(1/4)))] - p*PolyLog[2, (I*g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + I*d* 
g^(1/4))] - p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(I*e*(-f)^(1/4) + d*g^( 
1/4))] + p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4)) 
])/(2*Sqrt[-f]*Sqrt[g])
 

Rubi [A] (verified)

Time = 1.72 (sec) , antiderivative size = 543, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {2922, 2863, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx\)

\(\Big \downarrow \) 2922

\(\displaystyle 2 \int \frac {\sqrt {x} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{g x^2+f}d\sqrt {x}\)

\(\Big \downarrow \) 2863

\(\displaystyle 2 \int \left (-\frac {\sqrt {g} \sqrt {x} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{2 \sqrt {-f} \left (\sqrt {-f} \sqrt {g}-g x\right )}-\frac {\sqrt {g} \sqrt {x} \log \left (c \left (d+e \sqrt {x}\right )^p\right )}{2 \sqrt {-f} \left (g x+\sqrt {-f} \sqrt {g}\right )}\right )d\sqrt {x}\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 \left (-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt {-\sqrt {-f}}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}-\sqrt [4]{g} \sqrt {x}\right )}{d \sqrt [4]{g}+e \sqrt [4]{-f}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt {-\sqrt {-f}}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right ) \log \left (\frac {e \left (\sqrt [4]{-f}+\sqrt [4]{g} \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt {-\sqrt {-f}}-d \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,-\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{e \sqrt [4]{-f}-d \sqrt [4]{g}}\right )}{4 \sqrt {-f} \sqrt {g}}-\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt {-\sqrt {-f}}}\right )}{4 \sqrt {-f} \sqrt {g}}+\frac {p \operatorname {PolyLog}\left (2,\frac {\sqrt [4]{g} \left (d+e \sqrt {x}\right )}{\sqrt [4]{g} d+e \sqrt [4]{-f}}\right )}{4 \sqrt {-f} \sqrt {g}}\right )\)

Input:

Int[Log[c*(d + e*Sqrt[x])^p]/(f + g*x^2),x]
 

Output:

2*(-1/4*(Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[-f]] - g^(1/4)*Sqrt[x 
]))/(e*Sqrt[-Sqrt[-f]] + d*g^(1/4))])/(Sqrt[-f]*Sqrt[g]) + (Log[c*(d + e*S 
qrt[x])^p]*Log[(e*((-f)^(1/4) - g^(1/4)*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4 
))])/(4*Sqrt[-f]*Sqrt[g]) - (Log[c*(d + e*Sqrt[x])^p]*Log[(e*(Sqrt[-Sqrt[- 
f]] + g^(1/4)*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] - d*g^(1/4))])/(4*Sqrt[-f]*Sqrt 
[g]) + (Log[c*(d + e*Sqrt[x])^p]*Log[(e*((-f)^(1/4) + g^(1/4)*Sqrt[x]))/(e 
*(-f)^(1/4) - d*g^(1/4))])/(4*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, -((g^(1/4) 
*(d + e*Sqrt[x]))/(e*Sqrt[-Sqrt[-f]] - d*g^(1/4)))])/(4*Sqrt[-f]*Sqrt[g]) 
+ (p*PolyLog[2, -((g^(1/4)*(d + e*Sqrt[x]))/(e*(-f)^(1/4) - d*g^(1/4)))])/ 
(4*Sqrt[-f]*Sqrt[g]) - (p*PolyLog[2, (g^(1/4)*(d + e*Sqrt[x]))/(e*Sqrt[-Sq 
rt[-f]] + d*g^(1/4))])/(4*Sqrt[-f]*Sqrt[g]) + (p*PolyLog[2, (g^(1/4)*(d + 
e*Sqrt[x]))/(e*(-f)^(1/4) + d*g^(1/4))])/(4*Sqrt[-f]*Sqrt[g]))
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2863

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_)) 
^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(a 
 + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]
 

rule 2922

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*((f_) + 
 (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> With[{k = Denominator[n]}, Simp[k   S 
ubst[Int[x^(k - 1)*(f + g*x^(k*s))^r*(a + b*Log[c*(d + e*x^(k*n))^p])^q, x] 
, x, x^(1/k)], x] /; IntegerQ[k*s]] /; FreeQ[{a, b, c, d, e, f, g, n, p, q, 
 r, s}, x] && FractionQ[n]
 

Maple [F]

\[\int \frac {\ln \left (c \left (d +e \sqrt {x}\right )^{p}\right )}{g \,x^{2}+f}d x\]

Input:

int(ln(c*(d+e*x^(1/2))^p)/(g*x^2+f),x)
 

Output:

int(ln(c*(d+e*x^(1/2))^p)/(g*x^2+f),x)
 

Fricas [F]

\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \] Input:

integrate(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="fricas")
 

Output:

integral(log((e*sqrt(x) + d)^p*c)/(g*x^2 + f), x)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\text {Timed out} \] Input:

integrate(ln(c*(d+e*x**(1/2))**p)/(g*x**2+f),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \] Input:

integrate(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="maxima")
 

Output:

integrate(log((e*sqrt(x) + d)^p*c)/(g*x^2 + f), x)
 

Giac [F]

\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int { \frac {\log \left ({\left (e \sqrt {x} + d\right )}^{p} c\right )}{g x^{2} + f} \,d x } \] Input:

integrate(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x, algorithm="giac")
 

Output:

integrate(log((e*sqrt(x) + d)^p*c)/(g*x^2 + f), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int \frac {\ln \left (c\,{\left (d+e\,\sqrt {x}\right )}^p\right )}{g\,x^2+f} \,d x \] Input:

int(log(c*(d + e*x^(1/2))^p)/(f + g*x^2),x)
 

Output:

int(log(c*(d + e*x^(1/2))^p)/(f + g*x^2), x)
 

Reduce [F]

\[ \int \frac {\log \left (c \left (d+e \sqrt {x}\right )^p\right )}{f+g x^2} \, dx=\int \frac {\mathrm {log}\left (\left (\sqrt {x}\, e +d \right )^{p} c \right )}{g \,x^{2}+f}d x \] Input:

int(log(c*(d+e*x^(1/2))^p)/(g*x^2+f),x)
 

Output:

int(log((sqrt(x)*e + d)**p*c)/(f + g*x**2),x)